0.5x^2+3x=8

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Solution for 0.5x^2+3x=8 equation:


Simplifying
0.5x2 + 3x = 8

Reorder the terms:
3x + 0.5x2 = 8

Solving
3x + 0.5x2 = 8

Solving for variable 'x'.

Reorder the terms:
-8 + 3x + 0.5x2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 3x + 0.5x2 = 0

Begin completing the square.  Divide all terms by
0.5 the coefficient of the squared term: 

Divide each side by '0.5'.
-16 + 6x + x2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + 6x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + 6x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + 6x + x2 = 0 + 16
6x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
6x + x2 = 16

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = 16 + 9

Reorder the terms:
9 + 6x + x2 = 16 + 9

Combine like terms: 16 + 9 = 25
9 + 6x + x2 = 25

Factor a perfect square on the left side:
(x + 3)(x + 3) = 25

Calculate the square root of the right side: 5

Break this problem into two subproblems by setting 
(x + 3) equal to 5 and -5.

Subproblem 1

x + 3 = 5 Simplifying x + 3 = 5 Reorder the terms: 3 + x = 5 Solving 3 + x = 5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 5 + -3 Combine like terms: 3 + -3 = 0 0 + x = 5 + -3 x = 5 + -3 Combine like terms: 5 + -3 = 2 x = 2 Simplifying x = 2

Subproblem 2

x + 3 = -5 Simplifying x + 3 = -5 Reorder the terms: 3 + x = -5 Solving 3 + x = -5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -5 + -3 Combine like terms: 3 + -3 = 0 0 + x = -5 + -3 x = -5 + -3 Combine like terms: -5 + -3 = -8 x = -8 Simplifying x = -8

Solution

The solution to the problem is based on the solutions from the subproblems. x = {2, -8}

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